xrd indexing

8/10/2019 XRD Indexing

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Michael Carton

11352077

MTE 481: Analytical Methods for Materials

Lab 3: X-Ray Diffraction and Peak Indexing

Date of Experiment: 11/13/2014Submission Date: 12/01/2014

Instructor: Dr. Jinhui Song

TA: Chaolong Tang


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2

Abstract

The purpose of this experiment was to record the diffraction spectra for various samples

and to index their peaks using mathematical and analytical methods. These methods use a

combination of Braggs law and the plane spacing equations in order to determine the lattice

parameters.1A sample of nickel with a cubic structure and a sample of titanium with a hexagonal

structure were analyzed. The peak values, cubic structures, and lattice parameters determined by

the mathematical and analytical methods agree with each other and with the ICDD values

confirming that the materials indexed were FCC nickel and HCP titanium.


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3

Experimental Apparatus and Procedure

To begin the experiment a sample of pure nickel powder was acquired and placed in a

sample holder. The Ni powder and holder was loaded into the chamber of the Philips MPD XRD

machine. The doors were closed and secured and chiller checked to ensure that cool water was

running through the system. The operating computer was turned on and the Data Collector

program on the desktop was opened.

Once the program was opened, the software was connected to the hardware by clicking

Instrument then Connect and then OK. The Tension button was clicked and the voltage

and current were changed to 45kV/40mA. Test program 4 was run by selecting File

OpenProgramTest program 4. The start angle was set to 20, the end angle to 100, and the step

size to 0.05/step. The program was initiated by clicking Measure Program and giving the

file a unique name.

Once initiated, the shutter opened and scanned the sample according to the parameters set

in program 4. When the scan was complete, the shutter opened and the sample was removed

from the chamber. The data points from the scan were automatically saved by the computer and

exported to a .XRDML file.

To shut tool down, the voltage and current were reset to 30kV/10mA under the Tension

button on the program interface. The hardware was disconnected by clicking Instrument

Disconnect. The .XRDML file was converted to a .ASC file to be read by Excel and plotted.

The same procedure was repeated for the titanium powder. Standard XRD peak files

(ICDD) for both nickel and titanium samples were pulled for comparison. All of the peaks for

both samples were indexed using mathematical and analytical methods.


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4

Results and Discussion

Once the data was collected, it was plotted in Excel. The Nickel sample was analyzed

first and its peaks can be seen below in Figure 1.

Figure 1:Experimentally obtained XRD peaks for the Nickel sample

As seen above, there are five distinct peaks between the 2values of 20 and 100.

These values and their relative intensities are recorded in Table Ibelow.

Table I:Peaks from the Nickel sample and their relative intensitiesAngle Intensity Normalized Intensity

44.56 15630 10051.88 4820 30.876.38 1940 12.492.90 1380 8.8398.30 460 2.94

Next, the each of the peaks of the Nickel sample was indexed using both analytical and

mathematical methods. Following the methods learned in class, Tables II IVbelow were

constructed and used to determine the crystal structure and lattice parameter of the Nickel.


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Table II. Mathematical method for determining the lattice parameter and structure of a cubic material

Table III. Analytical method for determining the K value for a cubic material

K = 0.04784

Table IV.Analytical method for determining the lattice parameter and structure of a cubic material

PeakNo.

2 sin() sin()^21*sin()^2/sin(min)^2

2*sin()^2/sin(min)^2

3*sin()^2/sin(min)^2

(h^2)+(k^2)+(l^2) hkl a (angstroms)

1 44.56 0.38 0.14 1.00 2.00 3.00 3.00 111 3.52

2 51.88 0.44 0.19 1.33 2.66 3.99 4.00 200 3.52

3 76.38 0.62 0.38 2.66 5.32 7.98 8.00 220 3.52

4 92.90 0.72 0.53 3.65 7.31 10.96 11.00 311 3.52

5 98.30 0.76 0.57 3.98 7.96 11.94 12.00 222 3.53

Average: 3.52

PeakNo.

2 sin() sin()^2 sin()^2/2 sin()^2/3 sin()^2/4 sin()^2/5 sin()^2/6 sin()^2/7 sin()^2/8

1 44.56 0.37913 0.14374 0.07187 0.04791 0.03594 0.02875 0.02396 0.02053 0.01797

2 51.88 0.43743 0.19134 0.09567 0.06378 0.04784 0.03827 0.03189 0.02733 0.02392

3 76.38 0.61827 0.38226 0.19113 0.12742 0.09556 0.07645 0.06371 0.05461 0.047784 92.90 0.72477 0.52530 0.26265 0.17510 0.13132 0.10506 0.08755 0.07504 0.06566

5 98.30 0.75642 0.57218 0.28609 0.19073 0.14304 0.11444 0.09536 0.08174 0.07152

PeakNo.

2 (rad) sin()sin()^

2sin()^2/

K(h^2)+(k^2)+(l^2) hkl a (angstroms)

1 44.56 22.28 0.39 0.38 0.14 3.00 3.00 111 3.52

2 51.88 25.94 0.45 0.44 0.19 4.00 4.00 200 3.52

3 76.38 38.19 0.67 0.62 0.38 7.99 8.00 220 3.52

4 92.90 46.45 0.81 0.72 0.53 10.98 11.00 311 3.52

5 98.30 49.15 0.86 0.76 0.57 11.96 12.00 222 3.52


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These methods depend on the plane spacing equations and Braggs law in order to

determine the crystal structures and lattice parameters. Specifically, these equations can be

combined to make the relationship: 2()= (

4)(2+2+2). This equation can then be

solved either mathematically or analytically.2

As seen in Table II, the h2+k2+l2for each peak from the mathematical model are 3, 4, 8,

11, and 12. This indicates that the nickel has a face-centered cubic (FCC) structure.3The lattice

parameter was also calculated and found to be 3.52 .

Analytical methods were also used to confirm the crystal structure of the nickel. As seen

in Table IV, the h2+k2+l2for each peak from the analytical method are 3, 4, 8, 11, and 12. This

agrees with the mathematical model and also indicates that nickel has an FCC structure.3The

lattice parameter was once again calculated and found to be 3.52 which also agrees with the

mathematical model. The value of these lattice parameters are very close to the ICDD value for

nickel which is 3.54 . The ICDD data card is attached at the end of the report.

Overall, the peak angles, the FCC structure, and lattice parameter from both the

mathematical and analytical methods of indexing the peaks confirm that the material is in fact

nickel.


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Next, the experimental spectrum data for the titanium sample was plotted in Excel. This

can be seen below in Figure 2.

Figure 2: Experimentally obtained XRD peaks for the Titanium sample

As seen above, there are eleven clear peaks between the 2values of 20 and 100.

These values and their relative intensities are recorded in Table Vbelow.

Table V:Peaks from the titanium sample and their relative intensitiesAngle Intensity Normalized Intensity

34.96 625 22.538.30 835 30.140.06 2770 10052.86 380 13.762.78 380 13.770.52 390 14.176.08 360 13.077.20 290 10.582.16 90 3.2586.60 80 2.89

92.64 70 2.52

These peaks are very close to the expected ICDD peaks and were indexed using the

mathematical method in order to determine the crystal structure and lattice parameters. For

hexagonal structures, Braggs law can be combined with the plane spacing equations to get the


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8

following relationship: 2()=

4 [43 (

2++2] + (

). This relationship can then

be used to determine c and a, which are the lattice parameters.

First determine the possible values of 43[2++2]using all of the possibilities for h

and k. These values can be seen below in Table VI.

Table VI: Values for43 [

2++2]for all possible values of h and kk

0 1 2 3

0 0.000 1.333 5.333 12.000

h 1 1.333 4.000 9.333 17.333

2 5.333 9.333 16.000 25.333

3 12.000 17.333 25.333 36.000

Next, the possible values of

()

were determined for all possible values of l and using the

lattice parameter ratio (c/a) of 1.5871. These are below in Table VII.

Table VII: Values for

()

for all values of l and c/a = 1.5871

l l^2 l^2/(c/a)^2

0 0 0.0001 1 0.397

2 4 1.588

3 9 3.573

4 16 6.352

5 25 9.925

6 36 14.292

Then the solutions from Tables VI andVIIwere used to determine the43 [

2++

2] + (

)for every allowed hkl value. These were placed in order from smallest to largest in

Table VIII.


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Table VIII: All allowed hkl values and corresponding43[

2++2] + (

)values.

hkl value

100 1.333

002 1.588

101 1.73102 2.921

110 4.000

103 4.906

200 5.333

112 5.588

201 5.73

004 6.352

202 6.921

104 7.685

203 8.906210 9.333

211 9.73

114 10.352

212 10.921

105 11.258

204 11.685

300 12.000

213 12.906

302 13.588

006 14.292

205 15.258

106 16.625

This order of hkl values was then used to assign indices to the peaks from the diffraction

pattern. The avalues for the hk0type reflections was calculates and the cvalues for the 00ltype

reflections were calculated. These values were averaged in order to determine the a, c and c/a

values. This can all be seen below in Table IX.


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Table IX:Calculations of the c and a lattice parameters for the experimental spectrum

PeakNo.

2RelativeIntensity

sin()^2d

(nm)hkl a() c () (h^2)+hk+(k^2) l^2

1 34.96 22.5 0.0902 2.563 100 2.960 1

2 38.3 30.1 0.1076 2.347 002 4.695 43 40.06 100 0.1173 2.248 101

4 52.86 13.7 0.1981 1.730 102

5 62.78 13.7 0.2713 1.478 110 2.957 3

6 70.52 14.1 0.3333 1.334 103

7 76.08 13 0.3797 1.250 200 2.886 4

8 77.2 10.5 0.3892 1.234 112

9 82.16 3.25 0.4318 1.172 201

10 86.6 2.89 0.4703 1.123 004 4.491 16

11 92.64 2.52 0.5230 1.065 202

As seen in the table above the average values for a is 2.934 , the average value for c is

4.593 , and the average value for c/a is 1.565. These are close to the ICDD values for titanium

which are a = 2.951 , c = 4.670 , c/a = 1.583.

Summary and Conclusions

In conclusion, the experimental spectra of the nickel and titanium match the expected

spectra according to the ICDD data. The mathematical and analytical methods of indexing the

spectra agree with each other and result in crystal structures and lattice parameters that also agree

with the expected results. The experimental results indicate the nickel has an FCC structure and

the titanium has an HCP structure. The experimental value for the lattice parameter of nickel is

3.52 which is very close to the ICDD value of 3.54 . The experimental values for titaniums

lattice parameters are: a = 2.934 , c = 4.593 , c/a = 1.565 which is very close to the ICDD

values of: a = 2.951 , c = 4.670 , c/a = 1.583. The small variations can be accounted for by

the different systems used to generate the spectra and slight rounding errors.


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References

1. Callister, William D., Rethwisch, David G, Fundamentals of Materials Science andEngineering, 3rdEd., John Wiley and Sons, Inc., Hoboken, NJ, 2008

2. Suryanarayana, C.,Experimental Techniques in Materials and Mechanics, Taylor & FrancisGroup, Boca Raton, FL, 2011

3. Leng, Yang,Materials Characterization: Introduction to Microscopic and SpectroscopicMethods, John Wiley & Sons, Ltd., Hoboken NJ, 2008


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00-001-1258

Nov 13, 2014 5:56 PM (CAF User)

Status

Deleted QM:

Blank Pressure/Temperature:

Ambient ChemicalFormula:

Ni EmpiricalFormula:

Ni

Weight :

Ni100.00 Atomic :

Ni100.00 CompoundName:

Nickel

Radiation:

MoK : 0.7093

SYS: Cubic SPGR: Fm-3m (225)Author's Cell [ AuthCella: 3.54 AuthCell Vol: 44.36 AuthCellZ: 2.00 AuthCellMolVol: 22.18]

Density [Dcalc:

4.394g/cm Dmeas:

8.72g/cm] SS/FOM:

F(12) =7.9(0.117, 13)

Temp:

298.000K (Ambienttemperature assigned by ICDD editor) MeltingPoint:

1728 K

SpaceGroup:

Fm-3m (225) MolecularWeight:

58.70

CrystalData[XtlCella: 3.540 XtlCellb: 3.540 XtlCellc: 3.540 XtlCell : 90.00 XtlCell : 90.00XtlCell : 90.00 XtlCellVol: 44.36 XtlCellZ: 2.00] Crystal DataAxial Ratio [ a/b: 1.000 c/b: 1.000]ReducedCell [ RedCell a: 2.503 RedCellb: 2.503 RedCell c: 2.503 RedCell : 60.00

RedCell :

60.00 RedCell :

60.00 RedCellVol: 11.09]

Crystal (SymmetryAllowed):

Centrosymmetric

Pearson: cF2.00 Subfile(s): Common Phase, Forensic, Inorganic, Deleted Pattern, Metals & AlloysLastModificationDate: 01/11/2013 Cross-RefPDF s: 04-001-0091 (Alternate)

References:

Type

PrimaryReferenceOptical DataUnit Cell

DOI Reference

Hull. Phys. Rev. 17, 571 (1921).Data on Chem. for Cer. Use, Natl. Res. Council Bull. 107.The Structure of Crystals, 1st Ed.

DatabaseComments:

Color: White. Deleted Or Rejected By: Deleted by NBS card. Melting Point: 1728 K.

d-Spacings 8 - 00-001-1258 Fixed Slit Intensity - Cu K Avg 1.54184

2

44.407651.640376.1563

d

2.040000

1.770000

1.250000

I

1005040

h

122

k

102

l

100

* 2

92.189298.1913122.3379

d

1.070000

1.0200000.880000

I

60102

h

324

k

120

l

120

* 2

144.2592154.7640

d

0.8100000.790000

I

2016

h

34

k

32

l

10

*

Page 1 / 12014International CentreforDiffractionData.All rights reserved.


13/13

00-001-1197

Nov 13, 2014 3:29 PM (CAF User)

Status

Deleted QM:

Blank Pressure/Temperature:

Ambient ChemicalFormula:

Ti EmpiricalFormula:

Ti

Weight :

Ti100.00 Atomic :

Ti100.00 CompoundName:

Titanium

Radiation:

MoK : 0.7093

SYS: Hexagonal SPGR: P63/mmc (194)Author's Cell [ AuthCella: 2.951 AuthCell c: 4.67 AuthCell Vol: 35.22 AuthCellZ: 2.00AuthCellMolVol: 17.61] Author's Cell Axial Ratio [ c/a: 1.583]Density [Dcalc:

4.517g/cm Dmeas:

4.49g/cm] SS/FOM:

F(15) =5.1(0.162, 18)Temp: 298.000K (Ambienttemperature assigned by ICDD editor) MeltingPoint: 2093 K

SpaceGroup:

P63/mmc (194) MolecularWeight:

47.90CrystalData[XtlCella: 2.951 XtlCellb: 2.951 XtlCellc: 4.670 XtlCell : 90.00 XtlCell : 90.00XtlCell :

120.00 XtlCellVol: 35.22 XtlCellZ: 2.00]

CrystalDataAxial Ratio [ c/a:

1.583 a/b:

1.000 c/b:

1.583]

ReducedCell [ RedCell a: 2.951 RedCellb: 2.951 RedCell c: 4.670 RedCell : 90.00RedCell :

90.00 RedCell :

120.00 RedCellVol: 35.22]

Atomic parameters arecross-referencedfromPDFentry04-001-8963

Crystal (SymmetryAllowed): Centrosymmetric

SG SymmetryOperators:

Seq

1234

Operator

x,y,z-x,-y,-z-y,x-y,zy,-x+y,-z

Seq

5678

Operator

-x+y,-x,zx-y,x,-z-y,-x,zy,x,-z

Seq

9101112

Operator

x,x-y,z-x,-x+y,-z-x+y,y,zx-y,-y,-z

Seq

13141516

Operator

-x,-y,z+1/2x,y,-z+1/2y,-x+y,z+1/2-y,x-y,-z+1/2

Seq

17181920

Operator

x-y,x,z+1/2-x+y,-x,-z+1/2y,x,z+1/2-y,-x,-z+1/2

Seq

21222324

Operator

-x,-x+y,z+1/2x,x-y,-z+1/2x-y,-y,z+1/2-x+y,y,-z+1/2

Atomic Coordinates:

Atom

TiNum

1Wyckoff

2cSymmetry

-6m2x

0.33333y

0.66666z

0.25SOF

1.0IDP AET

12-d

Pearson: hP2.00 Subfile(s): Metals & Alloys, Inorganic, Deleted Pattern, Common Phase, Forensic, ExplosiveLastModificationDate: 01/11/2013 Cross-RefPDF s: 04-001-8963 (Primary)

References:

Type

PrimaryReferenceCrystal StructureOptical Data

Unit Cell

DOI Reference

Phys. Rev. 26, 56 (1925).Crystal Structure Source: LPF .Data on Chem. for Cer. Use, Natl. Res. Council Bull. 107.

The Structure of Crystals, 1st Ed.

DatabaseComments: Color: White. Deleted Or Rejected By: Deleted by NBS. Melting Point: 2093 K.

d-Spacings 15 - 00-001-1197 Fixed Slit Intensity - Cu K Avg 1.54184

2

35.052238.471240.261152.925862.7842

d

2.560000

2.3400002.240000

1.7300001.480000

I

40401004040

h

10111

k

00001

l

02120

* 2

70.243276.156377.623681.585186.0363

d

1.340000

1.2500001.2300001.1800001.130000

I

5040301010

h

11202

k

01000

l

32142

* 2

92.1892102.2847110.1952113.8499122.3379

d

1.0700000.9900000.9400000.9200000.880000

I

2030303010

h

12211

k

00110

l

43145

*

Page 1 / 12014International CentreforDiffractionData.All rights reserved.

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